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Xref: bloom-picayune.mit.edu rec.puzzles:18150 news.answers:3081 Newsgroups: rec.puzzles,news.answers Path: bloom-picayune.mit.edu!enterpoop.mit.edu!snorkelwacker.mit.edu!usc!sdd.hp.com!cs.utexas.edu!uunet!questrel!chris From: uunet!questrel!chris (Chris Cole) Subject: rec.puzzles FAQ, part 9 of 15 Message-ID: <puzzles-faq-9_717034101@questrel.com> Followup-To: rec.puzzles Summary: This posting contains a list of Frequently Asked Questions (and their answers). It should be read by anyone who wishes to post to the rec.puzzles newsgroup. Sender: chris@questrel.com (Chris Cole) Reply-To: uunet!questrel!faql-comment Organization: Questrel, Inc. References: <puzzles-faq-1_717034101@questrel.com> Date: Mon, 21 Sep 1992 00:09:26 GMT Approved: news-answers-request@MIT.Edu Expires: Sat, 3 Apr 1993 00:08:21 GMT Lines: 1553 Archive-name: puzzles-faq/part09 Last-modified: 1992/09/20 Version: 3 s Sunday School ss liner ss saints ss ship ss steamship st good man st hush st little way st paragon st road st saint st silence st stone st street st stumped st thoroughfare st way st weight stag speculator sten gun stet don't change it stir prison stop traffic signal sts saints sty filthy place stye eyesore su Soviet Union sub U-boat sub stand-in sub substitute sub warship supra over sure certain sw Cornwall sw Devon sw bridge opponents sw quarter sw south-west swift screecher swiss roll jammed cylinder sx Essex t Thailand t Tuesday t bandage t bar t bone t cart t cloth t cross t crossed t half dry t hundred and sixty t hundred and sixty thousand t junction t model + t peg t perfect letter t plate t rail t shirt t short time t square t tau t te t tea t tee t tesla t the t time t ton(ne) t tritium ta Territorial Army ta army ta cheers ta reserves ta soldiers ta terriers ta territorials ta thank you ta thanks ta volunteers tab label tace silence tag label tan beat tan brown tan maths function tar able seaman tar art nouveau tar sailor/salt/seaman tata Tosti's song tata goodbye tate gallery tau cross tay river tb torpedo boat td medal te Lawrence te note tea leaves tec detective ted Edward ted Heath tee peg teen old injury tees river tell archer temp secretary ten PM's address tene old injury tent wine ter three (triple) ter thrice test educational journal test examination test match teth Hebrew letter the article the articles - English ti note tic note tic spasm tic twitching tier row time father times daily timon misanthrope tin can tin cash tin money tin vessel tiny small tion empty container tit bird tit inferior horse tit poor horse tnt big banger tnt explosive tod fox todo commotion toe extremity toe member tom big bell tom cat tome book ton fashion ton hundred ton large amount ton weight tonne weight tor hell tor hill tor mountain tor point tor prominence tory Conservative tory party tory politician tp teepee tr Turkey tr transaction tr translation tram transport tree actor tres very (Fr.) tri three (triple) tri thrice troy ancient city troy old city try attempt try essay ts teas ts tees tt abstaining tt dry tt on the wagon tt race tt teas tt tees tt teetotal tt teetotaller tt thank you tu tradesmen tuck friar twelve eec two company u Conservative u Uruguay u Utah u about turn u acceptable u bend u boat u educational establishment u ewe u film u for all to see u high class u on view to all u posh u socially acceptable u suitable for children u superior u trap u tube u turn u union/Unionist u universal u university u upper class u uppish u upsilon u uranium u yew u you uc you see uk United Kingdom uk this country uk this island ule rubber ult last month um doubt um hesitation un United Nations un international un number one (Fr.) un one un one (dialect) un peacekeepers una number one (Ital.) unco very (Scot.) une number one (Fr.) uno international organisation uno number one (Ital.) up at university up excited up in court up mounted up riding up superior uq you queue ur ancient city ur hesitation ur old city ur primitive ur you are ure river uru Uruguay us America us American us as above us ewes us no good us transatlantic us undersecretary us use us useless us yews us you and me usa America use application use custom use employ(ment) use practice use practise ussr Soviet Union ut note ute half minute uu ewes uu use uu yews ux wife v Vatican v against v agent v bomb v day v five v look v neck v neckline v notch v opposing v see v sign v vanadium v vee v velocity v verb v verse v versus v very v victory v vide v volt v volume v win va Virginia vad nurse vale farewell vale goodbye vat tax vau Hebrew letter vb verb ve victory ver rev up very light vet surgeon vg for example vi half dozen vi six via old way vid see vid tanner/sixpence vide look vide see vin French wine vip big noise vip tanner/sixpence vir man/Roman vis viscount vj victory vo left hand vol book vol volume vy various years w Wednesday w Welsh w William w bridge players w direction w point w quarter w tungsten w watt w weak w west(ern) w whole numbers w wicket w width w wife w woman ward disadvantage (drawback) washington young feller we partnership we you and I wee little wee minor wee small who doctor wi Mayfair wi West Indies wi Westminster winner fabulous tortoise wise youth leaders wist knew (old word) women monstrous regiment woof bark wt small weight wt weight x Christ x PM's address x Xmas x across x body x chi x chromosome x cross x draw x ex,Exe x film x illiterate's signature x kiss x particle x ray x sign of love x sign of the times x spot marked x ten x ten thousand x thousand x times x unknown x vitamin x vote x wrong sign x xi xc ninety xi eleven xi side xi team xl excel xv side xv team y alloy y chromosome y level y measure y moth y one hundred and fifty y one hundred and fifty thousand y track y unknown y why y yard y year y yen y young y yttrium yard detectives yd measure ye the (old word) ye you (old word) yea agreement yew tree yr year yr your ys wise ys youth leaders yt that (old word) yu jade yule you will, say yy wise z Zambia z bar z bend z cedilla z final letter z integers z izzard z last character z last letter z omega z seven z seven thousand z sound of sleep z zed z zee z zero z zeta zo cross * zr Zaire zz (sound of) snoring ---------------------------------------------------------------------- -- Ross Beresford, | Email (trusted): rberesfo@cix.compulink.co.uk 10 Wagtail Close, | (work): ross@siesoft.co.uk Twyford, Reading, | (under test): ross@dickens.demon.co.uk RG10 9ED, UK | ==> games/crosswords/cryptic/double.p <== Each clue has two solutions, one for each diagram; one of the answers to 1ac. determines which solutions are for which diagram. All solutions are in Chamber's and Webster's Third except for one solution of each of 1dn, 3dn and 4dn, which can be found in Webster's 2nd. edition. ####################################################################### #1 |2 | | |3 |4 |5 #1 |2 | | |3 |4 |5 # # | | | | | | # | | | | | | # #----+----###########----#----#----#----+----###########----#----#----# #6 | |7 | | # # #6 | |7 | | # # # # | | | | # # # | | | | # # # #----#----#----######----#----#----#----#----#----######----#----#----# # # # #8 | | | # # # #8 | | | # # # # # | | | # # # # | | | # #----#----#----######----#----#----#----#----#----######----#----#----# #9 | | | # # # #9 | | | # # # # # | | | # # # # | | | # # # # #----#----#----######----#----#----#----#----#----######----#----#----# # # #10 | | | | # # #10 | | | | # # # # | | | | # # # | | | | # #----#----#----###########----+----#----#----#----###########----+----# #11 | | | | | | #11 | | | | | | # # | | | | | | # | | | | | | # ####################################################################### Ac. 1. What can have distinctive looking heads spaced about more prominently right. (7) 6. Vermin that can overrun fish and t'English tor perhaps. (5) 8. Old testament reversal - Adam's conclusion, start of sin. Felines initially with everything there. (4) 9. Black initiated cut, oozed out naturally. (4) 10. For instance, 11 with spleen dropping I count? (5) 11. Provoked explosion of grenade. (7) Dn. 1. Some of club taking part in theatrical function, for the equivalent of a fraction of a pound. (6) 2. Close-in light meter in one formation originally treated as limestone. (6) 3. Xingu River hombres having symmetrical shape. (5) 4. About sex-appeal measure - what waitresses should be? (6) 5. Old penny, least damaged, was preserved. (6) 7. IRA to harm ruling Englishman; extremes could be belonging to group. (5) ==> games/crosswords/cryptic/double.s <== +-+-+-+-+-+-+-+-+-+-+-+-+-+-+ |r e d c a p s|d e x t r a l| + + +-+-+ + + + + +-+-+ + + + |o t t e r|o|a|r o a c h|s|a| + + + +-+ + + + + + +-+ + + + |u|a|h|f a l l|a|z|m|t o m s| + + + +-+ + + + + + +-+ + + + |b l e d|r|i|t|c o o n|m|i|t| + + + +-+ + + + + + +-+ + + + |l|o|i r a t e|m|o|n o b l e| + + + +-+-+ + + + + +-+-+ + + |e n r a g e d|a n g e r e d| +-+-+-+-+-+-+-+-+-+-+-+-+-+-+ Notes. Left grid: Ac. 1. R + spaced (anag). 6. T'E tor (anag). 8. F-all. 9. B-led. 10. I-rate. Dn. 1. Ro-ub-le. 2. T.A.L. in one (anag). 4. it in pole. 5. anag of D+least. 7. anag of initial letters. Right grid: Ac. 1. D-extra-L. 6. 3 mngs. 8. OT (rev) + m-s. 9. initial letters. 10. No.-b(i)le. Dn. Dra-c-ma. 2. Zoo(m) in one (anag). 3. hidden. 4. SA (rev) + mile. 5. anag of D+least. 7. anag of final letters. -------------------------------------------------------------------- How I built it: it was hard! Basically, I started with a couple of word pairs which were easy to clue (e.g. enraged/angered - same meaning and anagrams of each other) and built a grid around them, trying to ensure corresponding words had something in common, either in meaning (their, among) or structure, (EtalON, EOzooN) and making sure that there was at least one word which could be used to distinguish the two grids (dextral). The clues were built in one of two ways: either the words had a common definition, and so a subsidiary indication which could refer to either was needed; or it was necessary to define each word in such a way that it was a subsidiary definition for all or part of the corresponding word, and deal with any remaining parts as before. I think the single hardest part was finding a definition of "interferometer" which could also be interpreted as "zoo" or "ozo". Roy rt@ukc.ac.uk ==> games/crosswords/cryptic/intro.p <== What are the rules for cluing cryptic crosswords? ==> games/crosswords/cryptic/intro.s <== This is a brief set of instructions for solving cryptic crossword puzzles for those of you who are intrigued by these puzzles, but haven't known how to begin solving them. For a more complete introduction, send a self-addressed, stamped envelope to The Atlantic Puzzler, 745 Boylston Street, Boston, Mass. 02116. The characteristic common to all cryptic crossword puzzles is the format of the clues. Each clue is a miniature word puzzle consisting of a straight definition of the answer and a cryptic definition of the answer. For example, Axle is poorly splined (7) yields SPINDLE. Axle is the straight definition. The cryptic definition (poorly splined) indicates an anagram of "splined". The number in parentheses is the number of letters in the answer. Punctuation and capitalization may be ignored in interpreting the clues. There are only eight categories of clues, as follows: 1. Anagram An anagram is a word formed by mixing up the letters of another word. An anagram clue is indicated by some word that means "mixed up", for example, out, crazy, bizarre, insane, etc. One or more words may contribute to the anagram. For example: Tim goes insane from selfishness (7) for EGOTISM (anagram of "Tim goes") 2. Double Definition A double definition is simply two definitions of the word. Most two-word clues are double definitions. For example: Release without charge (4) for FREE 3. Container A container clue indicates that something is to be put in (or wrapped around) something else. A container is indicated by phrases such as eaten by, contains, in, gobbles, etc. For example: In Missouri, consumed by fear (7) for AMONGST (MO = Missouri in ANGST = fear) 4. Hidden Word A hidden word is a word embedded in another word or words. It is indicated by phrases such as spot in, hides, at the heart of, covers, etc. For example: Worn spot in paper at typo (5) for RATTY (find ratty in "paper at typo") 5. Reversal A reversal is a definition of a word with the letters reversed. It is indicated by words such as back, reversed, up (for down clues), leftward (for across clues), etc. For example: Egad! Ray entirely reversed the lot of cloth (7) for YARDAGE ("Egad! Ray" reversed) 6. Homophone A homophone definition is a definition of a word that sounds the same as the answer, but is spelled differently. A homophone is indicated by words such as in audience, I hear, mouthed, verbally, etc. For example: Regrets prank, I hear (4) for RUES (the homophone is RUSE = prank) 7. Charade In a charade, the pieces of the word are "spelled" out in order. There are no auxiliary words that indicate a charade. For example: Excite a jerk extremist (7) for FANATIC (FAN = excite, A, TIC = jerk) 8. Deletion A deletion is a clue where you are instructed to remove a part of some word to make another word. For example, Times with poor wages (4) for AGES (with-poor WAGES, where with is abbreviated by W) Often the clue types are combined. Some common examples are 1) hidden word reversals where the answer is found backwards embedded in other words, and 2) containers or charades where the parts are anagrams. For example: Car shops have broken gear immersed in gasoline. (7) for GARAGES (RAGE = gear anagram in GAS = gasoline) All manner of common abbreviations, acronyms, and other symbology such as roman numerals are allowed. For example: c one hundred, cup, or centigrade vi six h hot s small ca california Two punctuation marks at the end of the clue have been reserved for special meaning. A question mark (?) indicates that the straight clue is not entirely straight (usually a pun). For example: I tie down mascara holder soundly? (7) for EYELASH (homophone of "I lash", mascara holder is a punning definition of EYELASH) An exclamation point (!) indicates that some part (usually all) of the clue overlaps. For example, the straight definition may also be the anagram indicator. Here is an example that entirely overlaps: A moped also has these! (6) for PEDALS (hidden word) Here, the entire clue indicates the hidden word, but the entire clue is also a straight definition of the answer. Give it a try! Cryptic crossword puzzles are a lot of fun. -- Steve Koehler ucsd.edu!telesoft!koehler telesoft!koehler@ucsd.edu koehler@telesoft.com ==> games/go-moku.p <== For a game of k in a row on an n x n board, for what values of k and n is there a win? Is (the largest such) k eventually constant or does it increase with n? ==> games/go-moku.s <== Berlekamp, Conway, and Guy's _Winning_Ways_ reports proof that the maximum k is between 4 and 7 inclusive, and it appears to be 5 or 6. They report: . 4-in-a-row is a draw on a 5x5 board (C. Y. Lee), but not on a 4x30 board (C. Lustenberger). . N-in-a-row is shown to be a draw on a NxN board for N>4, using a general pairing technique devised by A. W. Hales and R. I. Jewett. . 9-in-a-row is a draw even on an infinite board, a 1954 result of H. O. Pollak and C. E. Shannon. . More recently, the pseudonymous group T. G. L. Zetters showed that 8-in-a-row is a draw on an infinite board, and have made some progress on showing infinite 7-in-a-row to be a draw. Go-moku is 5-in-a-row played on a 19x19 go board. It is apparently a win for the first player, and so the Japanese have introduced several 'handicaps' for the first player (e.g., he must win with _exactly_ five: 6-in-a-row doesn't count), but apparently the game is still a win for the first player. None of these apparent results have been proven. ==> games/hi-q.p <== What is the quickest solution of the game Hi-Q (also called Solitair)? For those of you who aren't sure what the game looks like: 32 movable pegs ("+") are arranged on the following board such that only the middle position is empty ("-"). Just to be complete: the board consists of only these 33 positions. 1 2 3 4 5 6 7 1 + + + 2 + + + 3 + + + + + + + 4 + + + - + + + 5 + + + + + + + 6 + + + 7 + + + A piece moves on this board by jumping over one of its immediate neighboor (horizontally or vertically) into an empty space opposite. The peg that was jumped over, is hit and removed from the board. A move can contain multiple hits if you use the same peg to make the hits. You have to end with one peg exactly in the middle position (44). ==> games/hi-q.s <== 1: 46*44 2: 65*45 3: 57*55 4: 54*56 5: 52*54 6: 73*53 7: 43*63 8: 75*73*53 9: 35*55 10: 15*35 11: 23*43*63*65*45*25 12: 37*57*55*53 13: 31*33 14: 34*32 15: 51*31*33 16: 13*15*35 17: 36*34*32*52*54*34 18: 24*44 Found by Ernest Bergholt in 1912 and was proved to be minimal by John Beasley in 1964. References The Ins and Outs of Peg Solitaire John D Beasley Oxford U press, 1985 ISBN 0-19-853203-2 Winning Ways, Vol. 2, Ch. 23 Berlekamp, E.R. Academic Press, 1982 ISBN 01-12-091102-7 ==> games/jeopardy.p <== What are the highest, lowest, and most different scores contestants can achieve during a single game of Jeopardy? ==> games/jeopardy.s <== highest: $283,200.00, lowest: -$29,000.00, biggest difference: $309,700.00 (1) Our theoretical contestant has an itchy trigger finger, and rings in with an answer before either of his/her opponents. (2) The daily doubles (1 in the Jeopardy! round, 2 in the Double Jeopardy! round) all appear under an answer in the $100 or $200 rows. (3) All answers given by our contestant are (will be?) correct. Therefore: Round 1 (Jeopardy!): Max. score per category: $1500. For 6 categories - $100 for the DD, that's $8900. Our hero bets the farm and wins - score: $17,800. Round 2 (Double Jeopardy!): Max. score per category: $3000. Assume that the DDs are found last, in order. For 6 categories - $400 for both DDs, that's $17,600. Added to his/her winnings in Round 1, that's $35,400. After the 1st DD, where the whole thing is wagered, the contestant's score is $70,800. Then the whole amount is wagered again, yielding a total of $141,600. Round 3 (Final Jeopardy!): Our (very greedy! :) hero now bets the whole thing, to see just how much s/he can actually win. Assuming that his/her answer is right, the final amount would be $283,200. But the contestant can only take home $100,000; the rest is donated to charity. To calculate the lowest possible socre: -1500 x 6 = -9000 + 100 = -8900. On the Daily Double that appears in the 100 slot, you bet the maximum allowed, 500, and lose. So after the first round, you are at -9400. -3000 x 6 = -18000 + 400 = -17600 On the two Daily Doubles in the 200 slots, bet the maximum allowed, 1000. So after the second round you are at -9400 + -19600 = -29000. This is the lowest score you can achieve in Jeopardy before the Final Jeopardy round. The caveat here is that you *must* be the person sitting in the left-most seat (either a returning champion or the luckier of the three people who come in after a five-time champion "retires") at the beginning of the game, because otherwise you will not have control of the board when the first Daily Double comes along. ==> games/knight.tour.p <== For what board sizes is a knight's tour possible? ==> games/knight.tour.s <== A tour exists for boards of size 1x1, 3x4, 3xN with N >= 7, 4xN with N >= 5, and MxN with N >= M >= 5. In other words, for all rectangles except 1xN (excluding the trivial 1x1), 2xN, 3x3, 3x5, 3x6, 4x4. With the exception of 3x8 and 4xN, any even-sized board which allows a tour will also allow a closed (reentrant) tour. On an odd-sided board, there is one more square of one color than of the other. Every time a knight moves, it moves to a square of the other color than the one it is on. Therefore, on an odd-sided board, it must end the last move but one of the complete, reentrant tour on a square of the same color as that on which it started. It is then impossible to make the last move, for that move would end on a square of the same color as it begins on. Here is a solution for the 7x7 board (which is not reentrant). ------------------------------------ | 17 | 6 | 33 | 42 | 15 | 4 | 25 | ------------------------------------ | 32 | 47 | 16 | 5 | 26 | 35 | 14 | ------------------------------------ | 7 | 18 | 43 | 34 | 41 | 24 | 3 | ------------------------------------ | 46 | 31 | 48 | 27 | 44 | 13 | 36 | ------------------------------------ | 19 | 8 | 45 | 40 | 49 | 2 | 23 | ------------------------------------ | 30 | 39 | 10 | 21 | 28 | 37 | 12 | ------------------------------------ | 9 | 20 | 29 | 38 | 11 | 22 | 1 | ------------------------------------ Here is a solution for the 5x5 board (which is not reentrant). -------------------------- | 5 | 10 | 15 | 20 | 3 | -------------------------- | 16 | 21 | 4 | 9 | 14 | -------------------------- | 11 | 6 | 25 | 2 | 19 | -------------------------- | 22 | 17 | 8 | 13 | 24 | -------------------------- | 7 | 12 | 23 | 18 | 1 | -------------------------- Here is a reentrant 2x4x4 tour: 0 11 16 3 15 4 1 22 19 26 9 24 8 23 14 27 10 5 30 17 31 12 21 2 29 18 25 6 20 7 28 13 A reentrant 4x4x4 tour can be constructed by splicing two copies. It shouldn't be much more work now to completely solve the problem of which 3D rectangular boards allow tours. ==> games/nim.p <== Place 10 piles of 10 $1 bills in a row. A valid move is to reduce the last i>0 piles by the same amount j>0 for some i and j; a pile reduced to nothing is considered to have been removed. The loser is the player who picks up the last dollar, and they must forfeit half of what they picked up to the winner. 1) Who is the winner in Waldo Nim, the first or the second player? 2) How much more money than the loser can the winner obtain with best play on both parties? ==> games/nim.s <== For the particular game described we only need to consider positions for which the following condition holds for each pile: (number of bills in pile k) + k >= (number of piles) + 1 A GOOD position is defined as one in which this condition holds, with equality applying only to one pile P, and all piles following P having the same number of bills as P. ( So the initial position is GOOD, the special pile being the first. ) I now claim that if I leave you a GOOD position, and you make any move, I can move back to a GOOD position. Suppose there are n piles and the special pile is numbered (n-p+1) (so that the last p piles each contain p bills). (1) You take p bills from p or more piles; (a) If p = n, you have just taken the last bill and lost. (b) Otherwise I reduce pile (n-p) (which is now the last) to 1 bill. (2) You take p bills from r(<p) piles; I take r bills from (p-r) piles. (3) You take q(<p) bills from p or more piles; I take (p-q) bills from q piles. (4) You take q(<p) bills from r(<p) piles; (a) q+r>p; I take (p-q) bills from (q+r-p) piles (b) q+r<=p; I take (p-q) bills from (q+r) piles Verifying that each of the resulting positions is GOOD is tedious but straightforward. It is left as an exercise for the reader. -- RobH ==> games/othello.p <== How good are computers at Othello? ==> games/othello.s <== The interesting game in which computers are undoubted masters of all they survey is Othello, where Kai-Fu Lee's (CMU) program "Bill" is so good it can only play itself to learn to get better. Bill has a fantastically correct and efficient evaluation function, that recently has been further improved by learning coefficients for additional terms made up of the pair-wise combination of the four old terms. This improved the quality of the play approximately as much as searching an extra two ply. Bill is so good it can beat lots of players with no search at all. Its 6 or 7 ply search sweeps aside all opposition (though Kai-Fu says that some very good players are now coming along in Japan, and he is not sure whether Bill would beat them). One interesting question remaining in Othello is the game theoretic value of the starting position. Bill's results seem to indicate that the first player has an advantage. It appears that, since Kai-Fu has published all his evaluation material, someone could build an Othello machine, and produce a constructive proof (as was done for Cubic) that it is a win for the first player. ==> games/risk.p <== What are the odds when tossing dice in Risk? ==> games/risk.s <== Attacker using 3 dice, Defender using 2: Probability that Attacker wins 2 = 2323 / 7776 Probability that Attacker wins 1 = 3724 / 7776 Probability that Attacker wins 0 = 1729 / 7776 Attacker using 3 dice, Defender using 1: Probability that Attacker wins 1 = 855 / 1296 Probability that Attacker wins 0 = 441 / 1296 Attacker using 2 dice, Defender using 2: Probability that Attacker wins 2 = 225 / 1296 Probability that Attacker wins 1 = 630 / 1296 Probability that Attacker wins 0 = 441 / 1296 Attacker using 2 dice, Defender using 1: Probability that Attacker wins 1 = 125 / 216 Probability that Attacker wins 0 = 91 / 216 Attacker using 1 dice, Defender using 2: Probability that Attacker wins 1 = 90 / 216 Probability that Attacker wins 0 = 126 / 216 Attacker using 1 dice, Defender using 1: Probability that Attacker wins 1 = 15 / 36 Probability that Attacker wins 0 = 21 / 36 ==> games/rubiks.clock.p <== How do you quickly solve Rubik's clock? ==> games/rubiks.clock.s <== Solution to Rubik's Clock The solution to Rubik's Clock is very simple and the clock can be "worked" in 10-20 seconds once the solution is known. In this description of how to solve the clock I will describe the different clocks as if they were on a map (e.g. N,NE,E,SE,S,SW,W,NW); this leaves the middle clock which I will just call M. To work the Rubik's clock choose one side to start from; it does not matter from which side you start. Your initial goal will be to align the N,S,E,W and M clocks. Use the following algorithm to do this: [1] Start with all buttons in the OUT position. [2] Choose a N,S,E,W clock that does not already have the same time as M (i.e. not aligned with M). [3] Push in the closest two buttons to the clock you chose in [2]. [4] Using the knobs that are farest away from the clock you chose in [2] rotate the knob until M and the clock you chose are aligned. The time on the clocks at this point does not matter. [5] Go back to [1] until N,S,E,W and M are in alignment. [6] At this point N,S,E,W and M should all have the same time. Make sure all buttons are out and rotate any knob until N,S,E,W and M are pointing to 12 oclock. Now turn the puzzle over and repeat steps [1]-[6] for this side. DO NOT turn any knobs other than the ones described in [1]-[6]. If you have done this correctly then on both sides of the puzzle N,S,E,W and M will all be pointing to 12. Now to align NE,SE,SW,NW. To finish the puzzle you only need to work from one side. Choose a side and use the following algorithm to align the corners: [1] Start with all buttons OUT on the side you're working from. [2] Choose a corner that is not aligned. [3] Press the button closest to that corner in. [4] Using any knob except for that corner's knob rotate all the clocks until they are in line with the corner clock. (Here "all the clocks" means N,S,E,W,M and any other clock that you have already aligned) There is no need at this point to return the clocks to 12 although if it is less confusing you can. Remember to return all buttons to their up position before you do so. [5] Return to [1] until all clocks are aligned. [6] With all buttons up rotate all the clocks to 12. ==> games/rubiks.cube.p <== What is known about bounds on solving Rubik's cube? ==> games/rubiks.cube.s <== The "official" world record was set by Minh Thai at the 1982 World Championships in Budapest Hungary, with a time of 22.95 seconds. Keep in mind mathematicians provided standardized dislocation patterns for the cubes to be randomized as much as possible. The fastest cube solvers from 19 different countries had 3 attempts each to solve the cube as quickly as possible. Minh and several others have unofficially solved the cube in times between 16 and 19 seconds. However, Minh averages around 25 to 26 seconds after 10 trials, and by best average of ten trials is about 27 seconds (although it is usually higher). Consider that in the World Championships 19 of the world's fastest cube solvers each solved the cube 3 times and no one solved the cube in less than 20 seconds... God's algorithm is the name given to an as yet (as far as I know) undiscovered method to solve the rubik's cube in the least number of moves; as apposed to using 'canned' moves. The known lower bound is 18 moves. This is established by looking at things backwards: suppose we can solve a position in N moves. Then by running the solution backwards, we can also go from the solved position to the position we started with in N moves. Now we count how many sequences of N moves there are from the starting position, making certain that we don't turn the same face twice in a row: N=0: 1 (empty) sequence; N=1: 18 sequences (6 faces can be turned, each in 3 different ways) N=2: 18*15 sequences (take any sequence of length 1, then turn any of the five faces which is not the last face turned, in any of 3 different ways); N=3: 18*15*15 sequences (take any sequence of length 2, then turn any of the five faces which is not the last face turned, in any of 3 different ways); : : N=i: 18*15^(i-1) sequences. So there are only 1 + 18 + 18*15 + 18*15^2 + ... + 18*15^(n-1) sequences of moves of length n or less. This sequence sums to (18/14)*(15^n - 1) + 1. Trying particular values of n, we find that there are about 8.4 * 10^18 sequences of length 16 or less, and about 1.3 times 10^20 sequences of length 17 or less. Since there are 2^10 * 3^7 * 8! * 12!, or about 4.3 * 10^19, possible positions of the cube, we see that there simply aren't enough sequences of length 16 or less to reach every position from the starting position. So not every position can be solved in 16 or less moves - i.e. some positions require at least 17 moves. This can be improved to 18 moves by being a bit more careful about counting sequences which produce the same position. To do this, note that if you turn one face and then turn the opposite face, you get exactly the same result as if you'd done the two moves in the opposite order. When counting the number of essentially different sequences of N moves, therefore, we can split into two cases: (a) Last two moves were not of opposite faces. All such sequences can be obtained by taking a sequence of length N-1, choosing one of the 4 faces which is neither the face which was last turned nor the face opposite it, and choosing one of 3 possible ways to turn it. (If N=1, so that the sequence of length N-1 is empty and doesn't have a last move, we can choose any of the 6 faces.) (b) Last two moves were of opposite faces. All such sequences can be obtained by taking a sequence of length N-2, choosing one of the 2 opposite face pairs that doesn't include the last face turned, and turning each of the two faces in this pair (3*3 possibilities for how it was turned). (If N=2, so that the sequence of length N-2 is empty and doesn't have a last move, we can choose any of the 3 opposite face pairs.) This gives us a recurrence relation for the number X_N of sequences of length N: N=0: X_0 = 1 (the empty sequence) N=1: X_1 = 18 * X_0 = 18 N=2: X_2 = 12 * X_1 + 27 * X_0 = 243 N=3: X_3 = 12 * X_2 + 18 * X_1 = 3240 : : N=i: X_i = 12 * X_(i-1) + 18 * X_(i-2) If you do the calculations, you find that X_0 + X_1 + X_2 + ... + X_17 is about 2.0 * 10^19. So there are fewer essentially different sequences of moves of length 17 or less than there are positions of the cube, and so some positions require at least 18 moves. The upper bound of 50 moves is I believe due to Morwen Thistlethwaite, who developed a technique to solve the cube in a maximum of 50 moves. It involved a descent through a chain of subgroups of the full cube group, starting with the full cube group and ending with the trivial subgroup (i.e. the one containing the solved position only). Each stage involves a careful examination of the cube, essentially to work out which coset of the target subgroup it is in, followed by a table look-up to find a sequence to put it into that subgroup. Needless to say, it was not a fast technique! But it was fascinating to watch, because for the first three quarters or so of the solution, you couldn't really see anything happening - i.e. the position continued to appear random! If I remember correctly, one of the final subgroups in the chain was the subgroup generated by all the double twists of the faces - so near the end of the solution, you would suddenly notice that each face only had two colours on it. A few moves more and the solution was complete. Completely different from most cube solutions, in which you gradually see order return to chaos: with Morwen's solution, the order only re-appeared in the last 10-15 moves. With God's algorithm, of course, I would expect this effect to be even more pronounced: someone solving the cube with God's algorithm would probably look very much like a film of someone scrambling the cube, run in reverse! Finally, something I'd be curious to know in this context: consider the position in which every cubelet is in the right position, all the corner cubelets are in the correct orientation, and all the edge cubelets are "flipped" (i.e. the only change from the solved position is that every edge is flipped). What is the shortest sequence of moves known to get the cube into this position, or equivalently to solve it from this position? (I know of several sequences of 24 moves that do the trick.) The reason I'm interested in this particular position: it is the unique element of the centre of the cube group. As a consequence, I vaguely suspect (I'd hardly like to call it a conjecture :-) it may lie "opposite" the solved position in the cube graph - i.e. the graph with a vertex for each position of the cube and edges connecting positions that can be transformed into each other with a single move. If this is the case, then it is a good candidate to require the maximum possible number of moves in God's algorithm. -- David Seal dseal@armltd.co.uk To my knowledge, no one has ever demonstrated a specific cube position that takes 15 moves to solve. Furthermore, the lower bound is known to be greater than 15, due to a simple proof. The way we know the lower bound is by working backwards counting how many positions we can reach in a small number of moves from the solved position. If this is less than 43,252,003,274,489,856,000 (the total number of positions of Rubik's cube) then you need more than that number of moves to reach the other positions of the cube. Therefore, those positions will require more moves to solve. The answer depends on what we consider a move. There are three common definitions. The most restrictive is the QF metric, in which only a quarter-turn of a face is allowed as a single move. More common is the HF metric, in which a half-turn of a face is also counted as a single move. The most generous is the HS metric, in which a quarter- turn or half-turn of a central slice is also counted as a single move. These metrics are sometimes called the 12-generator, 18-generator, and 27-generator metrics, respectively, for the number of primitive moves. The definition does not affect which positions you can get to, or even how you get there, only how many moves we count for it. The answer is that even in the HS metric, the lower bound is 16, because at most 17,508,850,688,971,332,784 positions can be reached within 15 HS moves. In the HF metric, the lower bound is 18, because at most 19,973,266,111,335,481,264 positions can be reached within 17 HF moves. And in the QT metric, the lower bound is 21, because at most 39,812,499,178,877,773,072 positions can be reached within 20 QT moves. -- jjfink@skcla.monsanto.com writes: Lately in this conference I've noted several messages related to Rubik's Cube and Square 1. I've been an avid cube fanatic since 1981 and I've been gathering cube information since. Around Feb. 1990 I started to produce the Domain of the Cube Newsletter, which focuses on Rubik's Cube and all the cube variants produced to date. I include notes on unproduced prototype cubes which don't even exist, patent information, cube history (and prehistory), computer simulations of puzzles, etc. I'm up to the 4th issue. Anyways, if you're interested in other puzzles of the scramble by rotation type you may be interested in DOTC. It's available free to anyone interested. I am especially interested in contributing articles for the newsletter, e.g. ideas for new variants, God's Algorithm. Anyone ever write a Magic Dodecahedron simulation for a computer? Anyone understand Morwen Thistlethwaite's 50 move solution to Rubik's Cube? I'd love to hear from you. Drop me a SASE (say empire size) if you're interested in DOTC or if you would like to exchange notes on Rubik's Cube, Square 1 etc. I'm also interested in exchanging puzzle simulations, e.g. Rubik's Cube, Twisty Torus, NxNxN Simulations, etc, for Amiga and IBM computers. I've written several Rubik's Cube solving programs, and I'm trying to make the definitive puzzle solving engine. I'm also interested in AI programs for Rubik's Cube and the like. Ideal Toy put out the Rubik's Cube Newsletter, starting with issue #1 on May 1982. There were 4 issues in all, and I'm missing #3. I have: #1, May 1982 #2, Aug 1982 #3, Aug 1983 I am willing to trade photocopies with anyone to obtain #3. There was another sort of magazine, published in several languages called Rubik's Logic and Fantasy in space. I believe there were 8 issues in all. Unfortunately I don't have any of these! I'm willing to buy these off anyone interesting in selling. I would like to get the originals if at all possible... I'm also interested in buying any books on the cube or related puzzles. In particular I am _very_ interested in obtaining the following: Cube Games Don Taylor, Leanne Rylands Official Solution to Alexander's Star Adam Alexander The Amazing Pyraminx Dr. Ronald Turner-Smith The Winning Solution Minh Thai The Winning Solution to Rubik's Revenge Minh Thai Simple Solutions to Cubic Puzzles James G. Nourse I'm also interested in buying puzzles of the mechanical type. I'm still missing the Pyraminx Star (basically a Pyraminx with more tips on it), the Puck, and Hungarian Rings. If anyone out here is a fellow collector I'd like to hear from you. If you have a cube variant which you think is rare, or an idea for a cube variant we could swap notes. I'm in the middle of compiling an exhaustive library for computer simulations of puzzles. This includes simulations of all Uwe Meffert's puzzles which he prototyped but _never_ produced. In fact, I'm in the middle of working on a Pyraminx Hexagon solver. What? Never heard of it? Meffert did a lot of other puzzles which never were made. I invented some new "scramble by rotation" puzzles myself. My favourite creation is the Twisty Torus. It is a torus puzzle with segments (which slide around 360 degrees) with multiple rings around the circumference. The computer puzzle simulation library I'm forming will be described in depth in DOTC #4 (The Domain of the Cube Newsletter). So if you have any interesting computer puzzle programs please email me and tell me all about them! Also to the people interested in obtaining a subscription to DOTC, who are outside of Canada (which it seems is just about all of you!) please don't send U.S. or non-Canadian stamps (yeah, I know I said Self-Addressed Stamped Envelope before). Instead send me an international money order in Canadian funds for $6. I'll send you the first 4 issues (issue #4 is almost finished). Mark Longridge Address: 259 Thornton Rd N, Oshawa Ontario Canada, L1J 6T2 Email: mark.longridge@canrem.com One other thing, the six bucks is not for me to make any money. This is only to cover the cost of producing it and mailing it. I'm just trying to spread the word about DOTC and to encourage other mechanical puzzle lovers to share their ideas, books, programs and puzzles. Most of the programs I've written and/or collected are shareware for C64, Amiga and IBM. I have source for all my programs (all in C or Basic) and I am thinking of providing a disk with the 4th issue of DOTC. If the response is favourable I will continue to provide disks with DOTC. -- Mark Longridge <mark.longridge@canrem.com> writes: It may interest people to know that in the latest issue of "Cubism For Fun" % (# 28 that I just received yesterday) there is an article by Herbert Kociemba from Darmstadt. He describes a program that solves the cube. He states that until now he has found no configuration that required more than 21 turns to solve. He gives a 20 move manoeuvre to get at the "all edges flipped/ all corners twisted" position: DF^2U'B^2R^2B^2R^2LB'D'FD^2FB^2UF'RLU^2F' or in Varga's parlance: dofitabiribirilobadafodifobitofarolotifa Other things #28 contains are an analysis of Square 1, an article about triangular tilings by Martin Gardner, and a number of articles about other puzzles. -- % CFF is a newsletter published by the Dutch Cubusts Club NKC. Secretary: Anneke Treep Postbus 8295 6710 AG Ede The Netherlands Membership fee for 1992 is DFL 20 (about$ 11). -- -- dik t. winter <dik@cwi.nl> References: E. C. Turner & K. F. Gold, "Rubik's Groups", American Mathematical Monthly, vol. 92 (1985), pp. 617-629. Cubelike Puzzles - What Are They and How Do You Solve Them? J.A. Eidswick A.M.M. March, 1986 Rubik's Revenge: The Group Theoretical Solution Mogens Esrom Larsen A.M.M. June-July, 1985 The Group of the Hungarian Magic Cube Chris Rowley Proceedings of the First Western Austrialian Conference on Algebra, 1982 Rubik's Cubic Compendium Erno Rubik, Tamas Varga, et al (Ed by David Singmaster) Oxford University Press, 1987 (Some chapters on mathematics of the cube.) David Singmaster, _Notes on Rubik's `Magic Cube'_ "Winning Ways" by Berlekamp, Elwyn R. Conway, John H. Guy, Richard K. Volume two, pages 760-768, 808, 809 ==> games/rubiks.magic.p <== How do you solve Rubik's Magic? ==> games/rubiks.magic.s <== The solution is in a 3x3 grid with a corner missing. +---+---+---+ +---+---+---+---+ | 3 | 5 | 7 | | 1 | 3 | 5 | 7 | +---+---+---+ +---+---+---+---+ | 1 | 6 | 8 | | 2 | 4 | 6 | 8 | +---+---+---+ +---+---+---+---+ | 2 | 4 | Original Shape +---+---+ To get the 2x4 "standard" shape into this shape, follow this: 1. Lie it flat in front of you (4 going across). 2. Flip the pair (1,2) up and over on top of (3,4). 3. Flip the ONE square (2) up and over (1). [Note: if step 3 won't go, start over, but flip the entire original shape over (exposing the back).] 4. Flip the pair (2,4) up and over on top of (5,6). 5. Flip the pair (1,2) up and toward you on top of (blank,4). 6. Flip the ONE square (2) up and left on top of (1). 7. Flip the pair (2,4) up and toward you. Your puzzle won't be completely solved, but this is how to get the shape. Notice that 3,5,6,7,8 don't move. ==> games/scrabble.p <== What are some exceptional scrabble games? ==> games/scrabble.s <== The shortest scrabble game: The Scrabble Players News, Vol. XI No. 49, June 1983, contributed by Kyle Corbin of Raleigh, NC: [J] J U S S O X [X]U which can be done in 4 moves, JUS, SOX, [J]US, and [X]U. In SPN Vol. XI, No. 52, December 1983, Alan Frank presented what he claimed is the shortest game where no blanks are used, also four moves: C WUD CUKES DEY S This was followed in SPN, Vol. XII No. 54, April 1984, by Terry Davis of Glasgow, KY: V V O[X] [X]U, which is three moves. He noted that the use of two blanks prevents such plays as VOLVOX. Unfortunately, it doesn't prevent SONOVOX. - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Record for the highest scrabble score in a single turn (in a legal position): According to the Scrabble Players Newspaper (since renamed to Scrabble Players News) issue 44, p13, the highest score for one turn yet discovered, using the Official Scrabble Players Dictionary, 1st ed. (the 2nd edition is now in use in club and tournament play) and the Websters 9th New Collegiate Dictionary, was the following: d i s e q u i l i b r a t e D . . . . . . . e . . . . . . e . . . . . . . e . . . . . o m r a d i o a u t o g r a p(h)Y . . . . . . . . . . . w a s T . . . . . . . . . . b e . . h . . . . . . . . . . a . . g o . . . c o n j u n c t i v a L . . . . . . . . . . . . . n o . . . . . . . f i n i k i n G . . . . . . . a . . . (l) e i . . . . . . . d . s p e l t Z . . . . . . w e . . . . . . e . . . . . . r . . . . . . o r m e t h o x y f l u r a n e S for 1682 points. According to the May 1986 issue of GAMES, the highest known score achievable in one turn is 1,962 points. The word is BENZOXYCAMPHORS formed across the three triple-word scores on the bottom of the board. Apparently it was discovered by Darryl Francis, Ron Jerome, and Jeff Grant. As for other Scrabble trivia, the highest-scoring first move based on the Official Scrabble Players Dictionary is 120 points, with the words JUKEBOX, QUIZZED, SQUEEZE, or ZYMURGY. If Funk & Wagnall's New Standard Dictionary is used then ZYXOMMA, worth 130 points, can be formed. The highest-scoring game, based on Webster's Second and Third and on the Oxford English Dictionary, was devised by Ron Jerome and Ralph Beaman and totalled 4,142 points for the two players. The highest-scoring words in the game were BENZOXYCAMPHORS, VELVETEEN, and JACKPUDDINGHOOD. The following example of a SCRABBLE game produced a score of 2448 for one player and 1175 for the final word. It is taken from _Beyond Language_ (1967) by Dmitri Borgman (pp. 217-218). He credits this solution to Mrs. Josefa H. Byrne of San Francisco and implies that all words can be found in _Webster's Second Edition_. The two large words (multiplied by 27 as they span 3 triple word scores) are ZOOPSYCHOLOGIST (a psychologist who treats animals rather than humans) and PREJUDICATENESS (the condition or state of being decided beforehand). The asterisks (*) represent the blank tiles. (Please excuse any typo's). Board Player1 Player2 Z O O P S Y C H O L O G I S T ABILITY 76 ERI, YE 9 O N H A U R O W MAN, MI 10 EN 2 * R I B R O V E I FEN, FUN 14 MANIA 7 L T I K E G TABU 12 RIB 6 O L NEXT 11 AM 4 G I AX 9 END 6 I T IT, TIKE 10 LURE 6 * Y E LEND, LOGIC*AL 79 OO*LOGICAL 8 A R FUND, JUD 27 ATE, MA 7 L E N D M I ROVE 14 LO 2 E A Q DARE, DE 13 ES, ES, RE 6 W A X F E N U RE, ROW 14 IRE, IS, SO 7 E T A B U I A DARED, QUAD 22 ON 4 E N A M D A R E D WAX, WEE 27 WIG 9 P R E J U D I C A T E N E S S CHIT, HA 14 ON 2 PREJUDICATENESS, AN, MANIAC, QUADS, WEEP 911 OOP 8 ZOOPSYCHOLOGIST, HABILITY, TWIG, ZOOLOGICAL 1175 -------------------------------------- Total: 2438 93 F, N, V, T in loser's hand: +10 -10 -------------------------------------- Final Score: 2448 83 --------------------------------------------------------------------------- It is possible to form the following 14 7-letter OSPD words from the tiles: HUMANLY FATUOUS AMAZING EERIEST ROOFING TOILERS QUIXOTE JEWELRY CAPABLE PREVIEW BIDDERS HACKING OVATION DONATED ==> games/square-1.p <== Does anyone have any hints on how to solve the Square-1 puzzle?